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Math Expert V
Joined: 02 Sep 2009
Posts: 58404
Jamboree and GMAT Club Contest: In a class of 100 students, 80 passed  [#permalink]

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Difficulty:   95% (hard)

Question Stats: 41% (02:36) correct 59% (02:34) wrong based on 239 sessions

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Jamboree and GMAT Club Contest Starts

QUESTION #13:

In a class of 100 students, 80 passed Physics, 70 passed Chemistry, and 40 passed Math. If 10 students failed in all the three subjects, at least how many of the students passed all the three subjects?

(A) 0
(B) 5
(C) 10
(D) 20
(E) 25

Check conditions below:

For the following two weekends we'll be publishing 4 FRESH math questions and 4 FRESH verbal questions per weekend.

To participate, you will have to reply with your best answer/solution to the new questions that will be posted on Saturday and Sunday at 9 AM Pacific.
Then a week later, respective forum moderators will be selecting 2 winners who provided most correct answers to the questions, along with best solutions. Those winners will get 6-months access to GMAT Club Tests.

PLUS! Based on the answers and solutions for all the questions published during the project ONE user will be awarded with ONE Grand prize. He/She can opt for one of the following as a Grand Prize. It will be a choice for the winner:
-- GMAT Online Comprehensive (If the student wants an online GMAT preparation course)
-- GMAT Classroom Program (Only if he/she has a Jamboree center nearby and is willing to join the classroom program)

Bookmark this post to come back to this discussion for the question links - there will be 2 on Saturday and 2 on Sunday!

There is only one Grand prize and student can choose out of the above mentioned too options as per the conditions mentioned in blue font.
All announcements and winnings are final and no whining GMAT Club reserves the rights to modify the terms of this offer at any time.

NOTE: Test Prep Experts and Tutors are asked not to participate. We would like to have the members maximize their learning and problem solving process.

Thank you!

JAMBOBREE OFFICIAL SOLUTION

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Posts: 58404
Re: Jamboree and GMAT Club Contest: In a class of 100 students, 80 passed  [#permalink]

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Bunuel wrote:

Jamboree and GMAT Club Contest Starts

QUESTION #13:

In a class of 100 students, 80 passed Physics, 70 passed Chemistry, and 40 passed Math. If 10 students failed in all the three subjects, at least how many of the students passed all the three subjects?

(A) 0
(B) 5
(C) 10
(D) 20
(E) 25

Check conditions below:

For the following two weekends we'll be publishing 4 FRESH math questions and 4 FRESH verbal questions per weekend.

To participate, you will have to reply with your best answer/solution to the new questions that will be posted on Saturday and Sunday at 9 AM Pacific.
Then a week later, respective forum moderators will be selecting 2 winners who provided most correct answers to the questions, along with best solutions. Those winners will get 6-months access to GMAT Club Tests.

PLUS! Based on the answers and solutions for all the questions published during the project ONE user will be awarded with ONE Grand prize. He/She can opt for one of the following as a Grand Prize. It will be a choice for the winner:
-- GMAT Online Comprehensive (If the student wants an online GMAT preparation course)
-- GMAT Classroom Program (Only if he/she has a Jamboree center nearby and is willing to join the classroom program)

Bookmark this post to come back to this discussion for the question links - there will be 2 on Saturday and 2 on Sunday!

JAMBOBREE OFFICIAL SOLUTION:

There is only one Grand prize and student can choose out of the above mentioned too options as per the conditions mentioned in blue font.
All announcements and winnings are final and no whining GMAT Club reserves the rights to modify the terms of this offer at any time.

NOTE: Test Prep Experts and Tutors are asked not to participate. We would like to have the members maximize their learning and problem solving process.

Thank you!

JAMBOBREE OFFICIAL SOLUTION:

In the question on Venn diagrams when we have to calculate either the maximum or the minimum value as in this question ,we find the number line method one of the convenient methods

We will represent the total number of 100 students on the number line as shown we will start from the left to right and place the group with largest number .i.e physics in this case and the place the group with second largest number .i.e chemistry in this case from the right to left. Also it is mentioned that 10 students did not pass in any subject so we will keep them separate now we need to place the 40 students who passed the math subject among the 90 students as shown in the number line so that the over lap of all the three subjects the minimum. We can place 20 from 1 to 20 and 10 students from 80 to 90 but the remaining 10 students who passed math will been to be placed between 20 to 80 i.e they will be the students who have already passed physics and chemistry .so the minimum number of students who have passed all the three subjects is 10

Attachment: 1.png [ 588 Bytes | Viewed 37487 times ]

Attachment: 2.png [ 2.81 KiB | Viewed 37580 times ]

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Re: Jamboree and GMAT Club Contest: In a class of 100 students, 80 passed  [#permalink]

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Students that passed physics ,Set P=80
Students that passed chemistry , Set C = 70
Students that passed maths , Set M = 40

Students that failed all the 3 subjects = 10

Total = P + C + M - (PnC + CnM + PnM) + (PnCnM) + Neither
=>100 = 190 - (PnC + CnM + PnM) + (PnCnM) + 10
=> (PnCnM) = (PnC + CnM + PnM) - 100

For (PnCnM) to be minimum , (PnC + CnM + PnM) should be minimum .
So we need to the minimum overlap between the three sets.
For P and C ,
P=80
C=70
Then the total number of students that passed physcis and chemistry = 80+70 = 150
But there are only 90 students.
Therefore a minimum of 60 students passed physics and chemistry .
Similary for C and M ,
C+M=70+40=110
Therefore a minimum of 20 students passed Chemistry and Maths
And for P and M,
P+M=120
Therefore a minimum of 30 students passed Physics and Maths.
Mimimum value of (PnC + CnM + PnM) = 60+20+30=110

Minimum value of (PnCnM) = 110-100 =10

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Re: Jamboree and GMAT Club Contest: In a class of 100 students, 80 passed  [#permalink]

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In a class of 100 students, 80 passed Physics, 70 passed Chemistry, and 40 passed Math. If 10 students failed in all the three subjects, at least how many of the students passed all the three subjects?

(A) 0
(B) 5
(C) 10
(D) 20
(E) 25

Out of 100 students 10 failed in all 3 subjects. So, 90 students passed at least one of the subjects - Physics, Chemistry or Math.
We need to find the minimum value of the students passed in all 3 subs or the min. value of the overlapping of the 3 sets having - 80, 70 and 40 students, where the total sh be 90.
80 : 10
|-----------------------------------------------------|---------| 90

20 : 70
|---------------|-----------------------------------------------| 90

40 : 50
|----------------------------|----------------------------------| 90

The min. overlap of 80 and 70 is: 80+70-90 = 60.
The min. overlap of the common 60 and 40 is: 60+40-90 = 10.
This is the answer. C.
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Re: Jamboree and GMAT Club Contest: In a class of 100 students, 80 passed  [#permalink]

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In a class of 100 students, 80 passed Physics, 70 passed Chemistry, and 40 passed Math. If 10 students failed in all the three subjects, at least how many of the students passed all the three subjects?

Total = A + B + C - (Sum of 2 overlaps) - 2(all 3 overlaps) + Neither
100 = 80 + 70 + 40 - (Sum of 2 overlaps) - 2(all 3 overlaps) + 10
100 = 200 - (Sum of 2 overlaps) - 2(all 3 overlaps)

(Sum of 2 overlaps) + 2(all 3 overlaps) = 100 --- (1)
Also, max value of (Sum of 2 overlaps) + (all 3 overlaps) = 90 --- (2)
(1) - (2)
All 3 overlaps = 10.

Number of students who passed all 3 subjects = 10

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Re: Jamboree and GMAT Club Contest: In a class of 100 students, 80 passed  [#permalink]

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Please refer to the attached Picture for the solution

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Re: Jamboree and GMAT Club Contest: In a class of 100 students, 80 passed  [#permalink]

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Total students passed in atleast one subject = Total number of students - students failed in all 3 subjects
= 100 - 10 = 90
To find the least students passed in all 3 subjects (I used process eliminating each answer)
see attached venn diagrams
Attachment: sets.jpg [ 119.38 KiB | Viewed 29644 times ]

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Re: Jamboree and GMAT Club Contest: In a class of 100 students, 80 passed  [#permalink]

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1
Total Students = 100

Total passed = Total - failed = 100 - 10 = 90

Now the question is what is the minimum number of students passed in all three subjects?

Here our aim is to plug in such nummbers so that sum of total students passed is 90 and students passed in a subject individually remains what is given in question.

We notice, Choices A to E are in ascending order and we need the least number. so we will start plugging in the values from A to E.

Option - A, Passed in All three subjects = 0

1. Passed in Physics + Chemistry = 50
2. Passed in Chemistry + Maths = 20
3. Passed in Maths + Physics = 20
4. Passed in All three subjects (P,C,M) = 0

In this we notice that students passed in Physics = 1 + 3 + 4 = 50 + 20 + 0 = 90. But It is given that In physics only 80 passed.
Hence, "A" is wrong.

Option - B, Passed in All three subjects = 5

1. Passed in Physics + Chemistry = 50
2. Passed in Chemistry + Maths = 15
3. Passed in Maths + Physics = 20
4. Passed in All three subjects (P,C,M) = 5

In this we notice that students passed in Physics = 1 + 3 + 4 = 50 + 20 + 5 = 75. But It is given that In physics only 80 passed.
Hence, "B" is wrong.

Option - C, Passed in All three subjects = 10

1. Passed in Physics + Chemistry = 50
2. Passed in Chemistry + Maths = 10
3. Passed in Maths + Physics = 20
4. Passed in All three subjects (P,C,M) = 10

In this we notice that students passed in Physics, chemistry and maths are equal to given values and sum of all passed students is 90.
Hence, "C" is Correct.

We do not need to test D and E because question states "at least".
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My solution: ( C is the correct answer)

80 +70 + 40 - (numbers of those passed only 2 subjects (A) + 2* numbers of those passed 3 subjects (B)) = 100 - 10 =90
=> A+2*B = 100

Trials and errors: The question asks about the least possible value, so we start from the smallest.

(1) B = 0 => A = 100. But with 80, 70, and 40 students passed Physics, Chemistry, and Maths, the largest number of those passed only 2 subjects is 95 (max A =95) (80+15), as below => (1) INCORRECT

(2) B=5 => A=90. Since 5 students passed all 3 subjects, the remains are 75 Physics, 65 Chemistry, and 35 Maths. Same to (1), the max A = 87 (75+12), as below => (2) INCORRECT

(3) B=10 => A=80. Since 10 students passed all 3 subjects, the remains are 70 Physics, 60 Chemistry, and 30 Maths. Now it is possible to have 80 students passed only 2 subjects (70+10), as below So (C) is CORRECT

Originally posted by tronghieu1987 on 19 Nov 2015, 23:49.
Last edited by tronghieu1987 on 20 Nov 2015, 01:10, edited 2 times in total.
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Re: Jamboree and GMAT Club Contest: In a class of 100 students, 80 passed  [#permalink]

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In a class of 100 students, 80 passed Physics, 70 passed Chemistry, and 40 passed Math. If 10 students failed in all the three subjects, at least how many of the students passed all the three subjects?

Total Students = Physics (P) + Chemistry (C) + Maths (M) - (PnC+PnM+CnM) + (PnCnM) + Neither
100 = 80 + 70 + 40 - (PnC+PnM+CnM) + (PnCnM) + 10
(PnCnM) = (PnC+PnM+CnM) - 100..............(1)

Question: at least how many of the students passed all the three subjects?
Solution: Minimise (PnCnM)

So to minimise (PnCnM), we need to minimise (PnC+PnM+CnM), according to (1).

To get the overlap,

P+C=80+70=150
But the Total students are 100-10=90
So PnC could be 150-90=60

P+M=80+40=120
But total students are 90
So PnM could be 120-90=30

C+M=70+40=110
But the total students are 90
So CnM could be 110-90=20

Adding the overlaps we have 60+30+20=110 Or (PnC+PnM+CnM) = 110

So (PnCnM) = 110-100=10 (According to (1))

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Re: Jamboree and GMAT Club Contest: In a class of 100 students, 80 passed  [#permalink]

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1
QUESTION #13:

In a class of 100 students, 80 passed Physics, 70 passed Chemistry, and 40 passed Math. If 10 students failed in all the three subjects, at least how many of the students passed all the three subjects?

(A) 0
(B) 5
(C) 10
(D) 20
(E) 25

100 = P + C + M - (Sum of 2 subjects passed) - 2(all three subject passed) + Neither
100 = 80 + 70 + 40 - (2 subject) - 2(3 subject) + 10
(2 subject) + 2(3 subject) = 100

Also, passed in at least one subject is 90, i.e., passed in 1 subject + passed in 2 subject + passed in 3 subject = 90
To maximize pass in at least 2 subject, passed in 1 subject = 0, i.e., passed in 2 subject + passed in 3 subject = 90

From the above two equation, we can get passed in all three subject = 10.
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Re: Jamboree and GMAT Club Contest: In a class of 100 students, 80 passed  [#permalink]

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Total students = 100, Physics Passed = 80, Chemistry Passed = 70, Maths Passed = 40, All 3 failed =10.
Therefore Students who have not failed all three = 100 -10 =90
Out of these 90 students we have to find out at the least students who passed in all 3 subjects.

Out of 90 students , 80 have passed in Physics. So 10 have not passed in Physics.
Out of 90 students, 70 have passed in Chemistry. So 20 have not passed in Chemistry.
To minimise the number of students who have passed in all 3, lets take 30 students to have passed in 10 and 20 students who have failed in Physics and Chemistry respectively. So there are 10 more students left who have passed in Maths and that will be the overlap region students who have passed in all three subjects.

So 10 is the answer which is choice C.
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Re: Jamboree and GMAT Club Contest: In a class of 100 students, 80 passed  [#permalink]

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2
2
In a class of 100 students, 80 passed Physics, 70 passed Chemistry, and 40 passed Math. If 10 students failed in all the three subjects, at least how many of the students passed all the three subjects?

(A) 0
(B) 5
(C) 10
(D) 20
(E) 25

Ans:C ahh I marked the wrong one again ---I marked D in a hurry.
The answer can be arrived with the help of a formula--->
Total=A+B+C -(sum of exactly 2 groups overlap)- 2(All three) +Neither
100=80+70+40- (A)-2(B)+ 10
-100=-A-2B
100=A+2B
we need to find B. It is to found out at least how many students passed in all.
Sum of exactly two groups overlaps can be maximum 80 because it cannot be greater than the number of students passing in a single subject.
so, 100=80+2B
B=10
Will I get points for correct explanation, even my answer was wrong.
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Re: Jamboree and GMAT Club Contest: In a class of 100 students, 80 passed  [#permalink]

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Please refer the attachment for the solution. If any query is still there please reply.

Attachment:
File comment: In a class of 100 students, 80 passed FullSizeRender.jpg [ 949.84 KiB | Viewed 29318 times ]
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Re: Jamboree and GMAT Club Contest: In a class of 100 students, 80 passed  [#permalink]

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Now I understood it with your full explanation. I have more questions:

when I say min is 10, it means it may get higher than 10. what would be the case if 15 or 20? what would be the max number ? I think max = 40 students as I can't exceed number of students who passed in math.

Thanks
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Re: Jamboree and GMAT Club Contest: In a class of 100 students, 80 passed  [#permalink]

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Alternately we may also use the following method, which you all may find useful

We are here worried about the no. of students passed, so -
Total no. of students passed = 100-10=90 (Since 10 failed known from question)

Now since we need to find the min. students passed all 3
P&C = 70+80-90=150-90=60
(P&C)&M = 60+40-90=10

Therefore Ans: C (10)
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Re: Jamboree and GMAT Club Contest: In a class of 100 students, 80 passed  [#permalink]

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Bunuel wrote:

Jamboree and GMAT Club Contest Starts

QUESTION #13:

In a class of 100 students, 80 passed Physics, 70 passed Chemistry, and 40 passed Math. If 10 students failed in all the three subjects, at least how many of the students passed all the three subjects?

(A) 0
(B) 5
(C) 10
(D) 20
(E) 25

Check conditions below:

For the following two weekends we'll be publishing 4 FRESH math questions and 4 FRESH verbal questions per weekend.

To participate, you will have to reply with your best answer/solution to the new questions that will be posted on Saturday and Sunday at 9 AM Pacific.
Then a week later, respective forum moderators will be selecting 2 winners who provided most correct answers to the questions, along with best solutions. Those winners will get 6-months access to GMAT Club Tests.

PLUS! Based on the answers and solutions for all the questions published during the project ONE user will be awarded with ONE Grand prize. He/She can opt for one of the following as a Grand Prize. It will be a choice for the winner:
-- GMAT Online Comprehensive (If the student wants an online GMAT preparation course)
-- GMAT Classroom Program (Only if he/she has a Jamboree center nearby and is willing to join the classroom program)

Bookmark this post to come back to this discussion for the question links - there will be 2 on Saturday and 2 on Sunday!

There is only one Grand prize and student can choose out of the above mentioned too options as per the conditions mentioned in blue font.
All announcements and winnings are final and no whining GMAT Club reserves the rights to modify the terms of this offer at any time.

NOTE: Test Prep Experts and Tutors are asked not to participate. We would like to have the members maximize their learning and problem solving process.

Thank you!

Detailed explanation is as follows
Attachments

File comment: www.GMATinsight.com 121.jpg [ 66.42 KiB | Viewed 28695 times ]

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Re: Jamboree and GMAT Club Contest: In a class of 100 students, 80 passed  [#permalink]

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Vyshak wrote:
In a class of 100 students, 80 passed Physics, 70 passed Chemistry, and 40 passed Math. If 10 students failed in all the three subjects, at least how many of the students passed all the three subjects?

Total = A + B + C - (Sum of 2 overlaps) - 2(all 3 overlaps) + Neither
100 = 80 + 70 + 40 - (Sum of 2 overlaps) - 2(all 3 overlaps) + 10
100 = 200 - (Sum of 2 overlaps) - 2(all 3 overlaps)

(Sum of 2 overlaps) + 2(all 3 overlaps) = 100 --- (1)
Also, max value of (Sum of 2 overlaps) + (all 3 overlaps) = 90 --- (2)
(1) - (2)
All 3 overlaps = 10.

Number of students who passed all 3 subjects = 10

Can anyone please explain the area "Also, max value of (Sum of 2 overlaps) + (all 3 overlaps) = 90 --- (2)" How can we say this?
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Re: Jamboree and GMAT Club Contest: In a class of 100 students, 80 passed  [#permalink]

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buan15 wrote:
Vyshak wrote:
In a class of 100 students, 80 passed Physics, 70 passed Chemistry, and 40 passed Math. If 10 students failed in all the three subjects, at least how many of the students passed all the three subjects?

Total = A + B + C - (Sum of 2 overlaps) - 2(all 3 overlaps) + Neither
100 = 80 + 70 + 40 - (Sum of 2 overlaps) - 2(all 3 overlaps) + 10
100 = 200 - (Sum of 2 overlaps) - 2(all 3 overlaps)

(Sum of 2 overlaps) + 2(all 3 overlaps) = 100 --- (1)
Also, max value of (Sum of 2 overlaps) + (all 3 overlaps) = 90 --- (2)
(1) - (2)
All 3 overlaps = 10.

Number of students who passed all 3 subjects = 10

Can anyone please explain the area "Also, max value of (Sum of 2 overlaps) + (all 3 overlaps) = 90 --- (2)" How can we say this?

If you tend to get confused with formulas then refer the following explanation
Attachments

File comment: www.GMATinsight.com 121.jpg [ 66.42 KiB | Viewed 22811 times ]

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Re: Jamboree and GMAT Club Contest: In a class of 100 students, 80 passed  [#permalink]

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GMATinsight wrote:
buan15 wrote:
Vyshak wrote:
In a class of 100 students, 80 passed Physics, 70 passed Chemistry, and 40 passed Math. If 10 students failed in all the three subjects, at least how many of the students passed all the three subjects?

Total = A + B + C - (Sum of 2 overlaps) - 2(all 3 overlaps) + Neither
100 = 80 + 70 + 40 - (Sum of 2 overlaps) - 2(all 3 overlaps) + 10
100 = 200 - (Sum of 2 overlaps) - 2(all 3 overlaps)

(Sum of 2 overlaps) + 2(all 3 overlaps) = 100 --- (1)
Also, max value of (Sum of 2 overlaps) + (all 3 overlaps) = 90 --- (2)
(1) - (2)
All 3 overlaps = 10.

Number of students who passed all 3 subjects = 10

Can anyone please explain the area "Also, max value of (Sum of 2 overlaps) + (all 3 overlaps) = 90 --- (2)" How can we say this?

If you tend to get confused with formulas then refer the following explanation

I can understand the later part of the explanation but can't follow the first line i.e. 2(a+b+c)+d+e+f=80...
Where from can we derive the same?
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